I'd love to talk to set designers on these shows to find out what their influences were. I would not be at all surprised if these kinds of props originate in the Daisy Bell scene in 2001: A Space Odyssey, and I would bet any money that set in 2001 was an extrapolation of the SMS card.
Here's a "logic module" from ST:Enterprise - http://www.startrekpropauthority.com/2011/10/star-trek-enter... . It's apparently from "In A Mirror, Darkly", where the Terrans from the mirror universe acquired the Constitution-class Federation starship USS Defiant. So it's definitely an homage to the 1960s series.
I thought there was a similar style in Star Trek: The Original Series, but while they had card-loaded computers, it wasn't as an array of cards. Their hardware idiom is more like http://tosgraphics.yuku.com/topic/409/The-Type7-Console-Comp...
ST:TNG has an example at http://vignette3.wikia.nocookie.net/memoryalpha/images/b/bd/... but I suspect it was more directly derived from HAL than a 1960s era computer.